semisimple ring
A ring is (left) semisimple if it one of the following statements:
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1.
All left -modules are semisimple.
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2.
All finitely-generated (http://planetmath.org/FinitelyGeneratedRModule) left -modules are semisimple.
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3.
All cyclic left -modules are semisimple.
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4.
The left regular -module is semisimple.
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5.
All short exact sequences of left -modules split (http://planetmath.org/SplitShortExactSequence).
The last condition offers another homological characterization of a semisimple ring:
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A ring is (left) semisimple iff all of its left modules are projective (http://planetmath.org/ProjectiveModule).
A more ring-theorectic characterization of a (left) semisimple ring is:
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A ring is left semisimple iff it is semiprimitive and left artinian.
In some literature, a (left) semisimple ring is defined to be a ring that is semiprimitive without necessarily being (left) artinian. Such a ring (semiprimitive) is called Jacobson semisimple, or J-semisimple, to remind us of the fact that its Jacobson radical is (0).
Relating to von Neumann regular rings, one has:
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A ring is left semisimple iff it is von Neumann regular and left noetherian.
The famous Wedderburn-Artin Theorem that a (left) semisimple ring is isomorphic to a finite direct product of matrix rings over division rings.
The theorem implies that a left semisimplicity is synonymous with right semisimplicity, so that it is safe to drop the word left or right when referring to semisimple rings.
Title | semisimple ring |
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Canonical name | SemisimpleRing |
Date of creation | 2013-03-22 14:19:05 |
Last modified on | 2013-03-22 14:19:05 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 11 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 16D60 |
Related topic | SemiprimitiveRing |
Defines | semisimple |